I have a convergent function $f(x)$.
What I want to do is to create a table of values $f(x)$ until the difference in the two subsequent value satisfies some convergence criterion. Say, I want to stop if $f(x+1)-f(x)<0.1$. The function itself is very ugly and there are random components, so we cannot directly check when it passes the stopping criterion. How can I create a table of {f[1],f[2],\cdots,f[z]}, where z is the first number that passes the criterion.
As an example, let
f[x_] := LogisticSigmoid[x] init = 0.; step = 0.5; tol = 0.02; This method which relies on NestWhile[] is compact, but evaluates f repeatedly (which is fine if evaluating f is cheap):
Reap[NestWhile[# + step &, init, (f[#2] - Sow[f[#1]] > tol) &, 2]][[-1, 1]] (Of course, this can also be written using FixedPoint[]; I'll leave writing out that version as an exercise for the reader.)
Otherwise, we can use Sow[]+Reap[] in While[] if evaluating f can be expensive:
res = f[init]; Join[{res}, Reap[While[True, new = f[init += step]; If[new - res > tol, Sow[res = new], Break[]]]][[-1, 1]]]